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The University of Kansas science bulletin. 

[Lawrence] :University of Kansas, 1 902-1 996. 
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THE UNIVERSITY OF KANSAS 

SCIENCE BULLETIN 



Vol. XXXVIII, Pt. II] March 20, 1958 



[No. 22 



A Statistical Method for Evaluating Systematic 

Relationships 1 

BY 

Robert R. Sokal and Charles D. Michener 2 

Department of Entomology 
University of Kansas, Lawrence 

Abstract. Starting with correlation coefficients (based on numerous char- 
acters) among species of a systematic unit, the authors developed a method 
for grouping species, and regrouping the resultant assemblages, to form a classi- 
ficatory hierarchy most easily expressed as a treelike diagram of relationships. 
The details of the method are described, using as an example a group of bees. 
The resulting classification was similar to that previously established by classi- 
cal systematic methods, although some taxonomic changes were made in view 
of the new light thrown on relationships. The method is time consuming, al- 
though practical in isolated cases, with punched-card machines such as were 
used; it becomes generally practical with increasingly widely available digital 
computers. 

INTRODUCTION 

The purpose of the study reported here was to develop a quanti- 
tative index of relationship between any two species of a higher 
systematic unit, as well as to exploit such indices of association in 
the establishment of a satisfactory hierarchy. The authors became 
interested in the development of such a method when they at- 
tempted to find a technique for classifying organisms that was free 
from the subjectivity inherent in customary taxonomic procedure. 



1. Contribution number 945 from the Department of Entomology, University of Kansas. 

2. We wish to acknowledge the constructive criticism received in connection with this 
and related work from the following individuals who kindly gave their time to read and 
comment upon the manuscript: Paul R. Ehrlich, University of Kansas; Raymond B. Cat- 
tell, University of Illinois; Alfred E. Emerson, University of Chicago; Warwick E. Kerr, 
Universidade de Sao Paulo; Ernst Mayr, Harvard University; Louis L. McQuitty, Michigan 
State University; G. G. Simpson, American Museum of Natural History; Peter C. Silvester- 
Bradley, University of Kansas and University of Sheffield; and Paulo E. Vanzolini, De- 
partmento de Zoologia, Secretaria de Agricultura, Sao Paulo. These persons, however, are 
not responsible for the opinions which we have expressed. 

Acknowledgment is also due to the University of Kansas General Research Fund for 
assistance. 



( 1409 ) 



1410 The University Science Bulletin 



The systematic group chosen as a test of the feasibility of this under- 
taking was one consisting of 97 species of solitary bees in the family 




cgaehilidae. This choice was made because one of us (C. D. M.) 
has made recent systematic studies of these insects, so that conclu- 
sions as to the relationships obtained by the usual systematic pro- 
cedure could be compared with the results of the new method. 

The findings of our study as well as the philosophical bases of 
our attempts at quantifying systematic relationships have been re- 
ported elsewhere (Michener and Sokal, 1957). In this paper we 
propose to describe in some detail the actual method employed, 
as well as our reasons for adopting it and for rejecting several alter- 
nate procedures. It is our intention to illustrate the procedures in 
sufficient detail so that persons with a limited knowledge of statisti- 
cal methods will be able to follow our method. We expect our 
system to be applicable to most organisms, provided they exhibit a 
variety of characters, and the account to follow is consequently 
phrased in general terms. However, our practical illustrations are 
based on the bee group cited above in order to provide the reader 
with concrete examples. 

A quantitative method of finding the relationship between two 
species must be based on a number of taxonomic characters in a 
manner similar to the traditional systematic approach. However, 
whereas the latter technique generally uses few characters and 
weights these quite unequally and subjectively, the former method 
employs numerous but unweighted characters. Our reasons for not 
weighting characters have been detailed in the companion paper 
( Michener and Sokal, 1957). In the absence of an objective criterion 
of character weight it seems best to rely on a large number of 
equally weighted characters. In our bee study we employed 122 
characters per species; however, we feel significant results may be 
obtained from as few as 60 characters. 

Our use of the word "character" will require some elaboration. 
In its commonest taxonomic usage, a character is any feature of one 
kind of organism that differentiates it from another kind. Thus the 
red abdomen of one bee is a character distinguishing it from another 
bee with the abdomen black. In this paper we use the word in a 
second connotation only; that is, as a feature which varies from one 
kind of organism to another. Now, to use the above example, ab- 
dominal co'or is the character, which occurs in two "states" or alter- 
natives, red and black. 



Evaluating Systematic: Relationships 1411 



For each character the states were coded: 1, 2, 3, etc. In the 
bee study the number of states per character ranged from two to 
eight. Much variation in the number of states is undesirable from 
the point of view of the methods discussed below. In the study we 
undertook most characters had either three or four states. How- 
ever, when variation exceeds desirable bounds it might be prefer- 
able to divide the character state codes by a common denominator 
or to normalize them. 

The kinds of characters used in the bee study and the manner in 
which they were coded are discussed at length by Michener and 
Sokal ( 1957). The possible effect of parallelism is also treated in the 
same article. For purposes of the present paper the available data 
might be summarized as follows: we have records of a given num- 
ber (n) of species. For each species we have k records, k being 
the number of characters considered in the study. The coded values 
for any character may range from 1 to 9 depending on the number 



of states in which this character occurs in the group under con- 
sideration. As was mentioned previously it is desirable to have the 
number of states not differ too widely for the various characters. 
While it is not necessary to limit the number of possible character 
states to nine, our particular computational setup was greatly fa- 
cilitated by the use of a single digit code. 

PROCEDURES 

Character correlations and species correlations 

Two obvious ways suggested themselves to the authors regard- 
ing a procedure for deducing relationships from the character states 




a group of species. We could either correlate characters with 
each other or species with each other. Since both of these methods 
would lead to interpretable, although differing, results a brief dis- 
cussion of the implications of the two approaches follows. 

Sturtevant (1942) undertook a study of the genus Drosophila 
with objectives and procedures somewhat similar to ours. He re- 
corded 33 morphological, cytological and life history characters for 
each of 56 species of Drosophila and two species of the genus 



Scaptomyza. In his aim to develop a classification "as free from 
personal bias as I could make it," Sturtevant set up two tables. The 
first was a table of the total number of differences with respect to the 
33 chosen characters between any two of the 58 species. These give 
the degree of difference between the species concerned and are 



1412 The University Science Bulletin 




analogous to the complemental values of the "matching 
discussed in the section on Choice of a Correlation Coefficient be- 
low. 

A second table showed correlations between characters, expressed 
as two-way frequency distributions. By examining the three high- 
est character correlations Sturtevant found that six species con- 
sistently fell into the exceptional classes of the two-way frequency 
distributions. They were the two Scaptomyza species and four 
species of Drosophila which he thereupon placed in separate sub- 
genera. On the basis of the number of character differences be- 
tween and within subgenera Sturtevant was able to confirm this 
classification and arrive at some ideas on the relationships and ori- 
gins of the various groups. He also performed a similar analysis on 
29 characters of 40 genera of flies (Scatophaga, Conops and 38 as- 
sorted Acalypterae) to establish the relations of the family Droso- 
philidae. Unfortunately the paper cited lists only summaries of 
the 1 above tables and it is therefore difficult to compare Sturtevant's 
findings with ours. 

Correlation between characters ( R-technique in the idiom of the 
factor analysts) is the customary technique in biological and psy- 




gical studies involving correlational analysis. In character 



correlation matrices involving studies within one species each cor- 
relation represents the sum total of the common forces acting on 



any pair of characters. When analyzed by some method of factor 
analysis, the matrix customarily yields a so-called general size factor, 
a series of group factors affecting various groups of characters, and 
residual specific factors affecting single characters only. The fore- 
going is an example of a factor constellation involving morphologi- 
cal characters and is not necessarily the only possible constellation. 
As a matter of fact much psychometric work and the biometric 



papers by Howells (1951) and Stroud (1953) use the method of 



"simple structure" which a priori rejects solutions involving general 
factors. 

Regardless of the constellation preferred, the factors common to 
two characters and causing them to be correlated could be visualized 
as developmental forces, genetic or environmental in the final anal- 
ysis. The range of these genetic or environmental forces is de- 
pendent on the causes of variation within the sample of individuals 
studied. Thus a sample of individuals from an inbred, isogenic, line 
of animals would yield character correlations reflecting common 
nongenetic, physiological (i.e., caused by microecological dif- 



Evaluating Systematic Relationships 




ferences) factors only. Another sample comprising individuals from 
various races or subspecies would provide correlations based on 
common factors representing (1) genetic differences between in- 
dividuals; (2) genetic differences between races; (3) nongenetic 
physiological differences between individuals and (4) nongenetic 
ecological differences between races. One of the authors (R. R. S.) 
has been able to accumulate a series of character correlation mat- 
rices from various organisms representing these levels of variation. 
Matrices on correlation of aphid characters within galls (clones) 
and between galls have been published ( Sokal, 1952 ) while similar 
matrices on aphid correlations between localities and morphological 
correlations within and between strains of houseflies and Drosophila 
await suitable analysis and publication. 

When the sample transcends the bounds of the species the fac- 
tors behind a character correlation matrix take on new meaning: 
They now represent genetic divergence or the results of evolution- 
ary processes. In the one case they were ontogenetic forces, in the 
other they are phylogenetic forces. This type of analysis was 
pioneered by Stroud (1953) who analyzed correlations of 14 char- 
acters for soldiers of 48 species and imagines of 43 species of the 
termite genus Kalotermes. He was able to interpret some factors 
extracted from his correlation matrices as recognizable evolutionary 
trends. 

Another method of correlational analysis is called the transposed 
matrix method or the Q-technique (as compared with the R-tech- 
nique of character correlations, discussed above). 3 It consists of 
correlations between individuals based on measurements of char- 
acters which they have in common. In psychology this involves 
correlations between persons based on scores for common tests 
which these persons have taken. In the Q-technique we are in effect 
dealing with the same kind of raw data as in the R-technique, but 
we compute the correlation coefficients by summing squares and 



products at right angles to the direction previously taken (or we 
transpose the matrix before computation which amounts to the 
same thing). 

A Q-technique correlation coefficient in a study correlating in- 
dividuals of one species represents common forces or factors acting 
on the two individuals concerned. In this case we cannot speak of 
the "sum total of common forces" as we could in the case of the 



3. In a recent paper Cattell (1954) has suggested restricting the Q and R symbolism 
to studies involving factor analysis and proposed Q' and R' for studies, such as the present 
one, employing more superficial methods. 










The University Science Bulletin 



R-teehnique. Insofar as the characters used are indicative of the 
entire spectrum of potential variation of the individuals we can 
say that the resulting correlation coefficient is representative of the 
real affinity between two individuals. When scanned for clusters 
of high correlation coefficients the Q-type matrix reveals types of 
individuals which are similar. It is thus especially suited to classi- 
ficatory problems. When subjected to factor analysis the resulting 
factors are now of a different nature. The general size factor has 
been lost and in its place we find a general taxonomic group factor 
which accounts for the overall correlations of all the individuals in 
the study. 

When, as in the present study, the correlation is between species 
of a taxonomic unit the general factor is a general systematic factor 
denoting overall relationship within the systematic group. The 
species having the highest factor loading would be most representa- 
tive of the group. Other factors would describe subgroups within 
the systematic unit and describe the relationships of these subgroups 
with each other and of the species to the subgroups. It should be 
clear from the above that for purposes of biological classification 
the relationships represented by a Q-technique matrix are more 
meaningful by far than are those of a R-technique matrix. Except 
for the above-mentioned work of Sturtevant (1942) which involved 
not correlations but character differences, the only Q-type study 
in systematics of which the authors are aware is in a publication 
by one of them ( Sokal, 1958 ) containing factor analyses of selected 
portions of the present data. A number of the phytosociological 
coefficients of association and similarity can be considered as of the 
Q-type. 

Psychologists have used Q-technique repeatedly {e. g., Burt 1937, 
Stephenson 1936), although R-technique is still preferred in most 
studies. Cattell (1952) has listed 5 points of criticism of the Q- 
nique. It is appropriate that we discuss briefly their relation to 
the problems under study here. The first objection is that Q-tech- 
nique loses the general size factor, yielding in its place a common 




species factor. This latter is claimed to be trivial by Cattell, and 
correctly so, for psychological work. However, in a matrix of cor- 
relations between species such a general systematic factor delineates 
the relation of individual species to the taxonomic group and in- 
dicates the proportion of the variance of each species explained by 
the general systematic factor. 

Cattell's second objection to Q-technique is that it is unreasonable 
to assume simple structure in the factorization of a Q-matrix. The 



Evaluating Systematic Relationships 1415 

authors agree with this argument, but for the purposes of the pres- 
ent paper it is not important since they are not here undertaking a 
factor analysis. Furthermore, they feel that simple structure ( i. e., 



S 



is 




not necessarily a very suitable constellation for many biological 
factorizations. 

e third objection refers to a customary shortcoming of Q- 
matrices. They are based on few individuals and generalizations 
about the entire population are drawn from them. In this study, 
the matrix is of course of more than adequate size. Furthermore 
our conclusions are not intended to extend to species not included 
in our study. 

It is true that the species recorded are an eclectic sample from 
those extant in the world today. On the other hand we are of course 
dealing with a sample obtained by natural selection from the multi- 
tude of species or specieslike entities that have existed since the 
origin of the four genera of this study. Hypotheses regarding these 
extinct species will be valid only insofar as recent species reflect the 
course of evolutionary history. 

Another point in connection with the third objection is the num- 



ber of characters employed. True relationships will become 
apparent only insofar as the characters adequately represent the 
sources of variation within the species. 

A fourth objection relates to the lack of equivalence in recording 
and interpreting the factors from the Q- and R-matrices. It com- 
pares the relative permanence of psychological tests with the rela- 
tive impermanence of persons. In this study we are confronted 
with characters and species varying in their relative permanence, 
but both equally permanent when based on the time scale of the 
scientist investigating them. 

The fifth criticism, labelling the Q-technique as descriptive rather 
than predictive, again is invalid when applied to the present data. 
Since the purpose of the study is historically descriptive and one of 
our aims is to divide the population of species into categories, the 
technique's fault for psychological research becomes a virtue in our 
field of investigation. 

There are two evolutionary situations under which it is important 
to examine the two types of matrices. The first might be referred 
to as breakage of correlation. It occurs when in a certain evolution- 
ary line two characters that were correlated in ancestral lines and 
are still correlated in related lines become independent of each 









13_8050 



1416 The University Science Bulletin 



other. 



g 



onditions the R-matrix is a poor representation 
>etween the two characters. There is no good 
such a correlation, close in one line, absent in 
the other. On the other hand a Q-matrix is not affected by such 

data. 

Convergence of species for a number of characters is a second 
disturbing phenomenon. Here the R-matrix is not affected while 
the Q-matrix is affected if the convergent characters outweigh the 
nonconvergent ones in numbers. 

We do not believe this is likely if an adequate number of char- 
acters is studied. In case of a preponderance of convergent char- 
acters and in the absence of paleontological data it is doubtful 
whether the systematists would be able to distinguish convergence 
from relationship by descent. 

From a consideration of the above arguments it follows that given 
the objectives and material of the present study the Q-technique is 
to be preferred to the R-technique and the objections made by 
Cattell to the former method do not apply to our case. However, 
besides the theoretical reason for adopting the Q-technique as re- 
flecting relationships between species there were several practical 
reasons for so doing. The problem of finding a suitable type of 
correlation coefficient between characters would have been formid- 
able in view of the coding system adopted. Since some of the char- 
acters were present in two states only while others were present in 
as many as eight states, there would probably not have been any 
one type of correlation coefficient for all possible character com- 
binations. A matrix based on correlation coefficients of different 
types would be far from desirable. Furthermore, uniformity of 
computational procedure was essential to efficient handling of the 
data by International Business Machines (IBM) equipment. 

Not to be underestimated is the saving in computation resulting 
from adoption of a 97 x 97 species correlation matrix vs. a 122 x 122 
character correlation matrix. The former requires the computation 
of only 4656 correlation coefficients while the latter would neces- 
sitate 7381 such coefficients. 

The choice of a correlation coefficient 

As a next step a suitable correlation coefficient had to be chosen 
to represent the correlations between species. There were serious 
considerations against the use of the product-moment correlation 
coefficient since the variables (species) are anything but normally 
distributed. Table 1 presents frequency distributions of state codes 



Evaluating Systematic Relationships 



1417 



Table 1 
Frequency distributions of state codes for the characters of species 19, 56, 



83 and 84. 



State 
code 



Sp. 19 
f 



Sp. 56 
f 



Sp. 83 
f 



Sp. 84 



1 

2 

3 
4 
5 
6 

7 



54 



°1 



31 



3 
2 



56 
40 
14 
11 
1 



48 

42 

23 

7 

2 



46 
41 

26 
6 
2 
1 



1 



Sf 



122 



122 



122 



122 



for four representative species. The distributions are highly asym- 
metrical. Those for species 19 and 56 approach Poisson distribu- 
tions for their means when the class codes are reduced by one. 
Any interpretation of this agreement is dubious, however, in view 
of the variable number of states possible per character. 

Other correlation coefficients were considered and rejected. The 
correlation ratio, -q, is unsuitable since ^ y does not necessarily equal 

Tetrachoric r would have lost some of the information avail- 
able because it would necessitate reducing all characters to two 



n g 



yVx 




Furthermore the theoretical 




of 




e 




normality essential to correct application of the tetrachoric correla- 
tion coefficient cannot be defended for all characters. 

Another method of demonstrating an association between species 
would be the very simple one of counting the numbers of matches 
in states for the 122 characters of any pair of species of bees and 
then dividing this number by 122, the highest possible number of 
such matches. The results for species 19, 56, 83, and 84 are shown 
on table 2 where these "matching coefficients" are compared with 
product-moment correlation coefficients. The "matching coeffici- 
ents" are somewhat higher than the correlation coefficients but 
resemble them in relative magnitude. In spite of this fact, "match- 

Table 2 

"Matching coefficients" (below diagonal) and product-moment correlation 
coefficients (above diagonal) between species 19, 56, 83 and 84. 



19 
56 
83 
84 



19 

X 

.52 
.53 
.50 




^—^—^^— 



.40 

X 

.61 

.54 



83 

.37 

.47 

X 

.87 



84 

.37 

.38 

.93 

X 



1418 The University Science Bulletin 



ing coefficients" were not used since they have an unknown sampling 
distribution, they distort resemblances by counting a 3 to 4 mis- 
match the equal of a 1 to 7 mismatch, and finally they would 
have been harder to handle by the IBM equipment available to us. 
Lacking a more suitable means of correlation we adopted the 
product-moment r, in spite of nonnormal distribution of variates 
and possible heteroscedasticity. Various ways of improving the 
distributions by means of transformations were tried. Table 3 shows 
the same correlation coefficients as the upper half of the matrix of 

— ■ 

2, but based on \/ X and \/ X +-5 transformations. The 
slight differences obtained do not justify the extra computational 
labor involved. 

We have already briefly touched on the desirability of coding the 
data in such a way as to put all character states on the same scale. 
In a character with two states the code 2 indicates a situation dif- 




g 



Q 



the scores for different tests are often not in comparable units. This 
situation is usually met by normalizing the rows (tests, or in our 
case characters ) of the raw score matrix. The authors did not per- 

Table 3 

Product-moment correlation coefficients between species 19, 56, 83 and 84 
based on variates coded as V X ( below diagonal ) and as V X + -5 ( above 
diagonal). Compare with uncoded product-moment correlation coefficients in 
table 2. 



19 56 83 84 



19 

56 
83 
84 



X .42 .36 .37 

.42 X .50 .41 

.36 .51 X .93 

■ 

.37 .41 .93 X 



form this transformation since ( 1 ) it would have removed the com- 
mon systematic factor from the matrix of correlations and would 
thus have lowered the correlation coefficients considerably; (2) ap- 
plication of the character state codes does standardize the data to 
a certain extent because 76 percent of the characters have either 
three or four states and only 3 percent have six or more states; (3) 
although the additional labor of normalizing the variates would not 
have been excessive the amount of IBM work involved in comput- 
ing correlation coefficients would have been prohibitive, since a 
one-digit code would not have sufficed for normalized data. 

The authors are well aware that their methodology of coding and 
correlation could profit by refinement. It is, however, our point of 



Evaluating Systematic Relationships 





view that in a pilot study of this nature such refinements are pre- 

Should the general method prove of value, significant re- 
sults will surely emerge in spite of minor imperfections in technique. 

Compulation 

The computation of a large matrix of correlation coefficients such 
as the 97 x 97 bee matrix presents serious technical difficulties. Only 
high speed electronic computing machines are able to perform this 
operation with real dispatch. At the time our bee data were being 
processed we had only punched-card tabulating machines at our 
disposal. It might be noted here that a computational operation of 
this magnitude cannot reasonably be undertaken without some auto- 
matic computing facilities. The equipment used by the authors is 



that available in the University of Kansas IBM laboratory: a card 
punch (type 26), a verifier (type 56), an accounting machine (type 
402) and a reproducing machine (type 514). 

The computational problem was simplified somewhat by the fact 
that the variates consisted of single digits only. This increased the 
number of variables that the machine could process simultaneously. 
Each IBM card represented a character with the state code of each 
species for the particular character listed in separate columns. Since 
there are only 80 columns per card, it was impossible to record all 
species on any one card. A different approach was therefore 
adopted and the card divided as follows: 

Column 1 — Project code 

Columns 2-4 — Character code number 

Column 5 — Deck code (explained below) 

Columns 6-8 — Left blank for possible subsequent use 

Columns 9-44 — Multiplier columns for 36 species 



Columns 45-80 — Multiplicand columns for 36 species. 
The 97 species were divided into group I for species 1-36, group II 
for species 37-72 and group III for species 73-97. Since group III 
used only 25 columns another 5 columns were taken up by a repeti- 
tion of data on species 1 through 5, which we used as a check on 
computational procedure. Six decks of 122 cards each, one card 





were then prepared. 



tuted 



follows : 



Deck Multiplier Multiplicand 

1 Group I Group I 

2 Group II Group II 

3 ' Group III Group HI 

4 Group I Group II 

5 Group II Group III 

6 Group I Group III 



1420 The University Science Bulletin 



Different card colors besides a punched code were used to dis- 
tinguish the decks. 

By running these decks in succession through the tabulator we 
were able to reduce rewiring of the board to one half of what it 
would have been with the minimum number of decks (3). 

The method of arriving at the 2x 2 and Sxy was the customary 
one of progressive digiting with interspersed "X-cards." Running 
time on the 402 tabulator was some 24 hours. Punching and verify- 
ing of the cards had taken a similar amount of time. Thus the 
preparation of the 2x 2 and 2xy for the entire matrix took about a 
week. These values were computed for a half-matrix only. How- 
ever, a test deck and five test variables detected wiring errors and 
machine malfunction with a reasonable limit of safety. 

The next step was the computation of the correlation coefficients. 
This was done by computers using desk calculators. 4 The matrix of 
squares and products was subdivided into manageable sections, 30 



variables (species) square. All computations were checked by a 
different computer and, where possible, by different steps. The 
computational procedure employed was the customary L method. 5 
It does not seem necessary to elaborate on the details of this method. 
Any good textbook of statistics will contain a section on the com- 
putation of a product-moment correlation coefficient. Furthermore, 
each computation center has its own setup for correlation coefficients 
depending on the capabilities of the machines and thus no general 
account need be presented here. 



The correlation coefficients were calculated to four significant 
decimal places and entered on a matrix. Three decimal places 
would have been quite sufficient for this study; however four were 
computed in case later statistical work required greater refinement. 
Total computation time for this phase of the work was 160 man- 
hours. It should be emphasized that the time estimates given above 
refer to the relatively simple equipment available to us. Digital 
computers are now available which would handle the entire com- 
putation, from raw data to completed correlation matrix without 
human intervention in less than an hour. This would be only one- 
two hundredths of the time it took us to compute the same informa- 



4. The writers at this point wish to express their appreciation to Misses Betty Becker, 
Marion Clyma, Jacqueline Johnson, Normandie Morrison, and Messrs. D. A. Crossley, Jr., 
Ralph Jones and Roger Price for their conscientious assistance with IBM work and desk 
computation. 



5. r xy = L xy / V L x VLy> where L xv = N2XY — 2X2 Y and 



L x = N2X 2 — (2X) 2 , L y = N2Y 2 — (2Y) 2 . 



Evaluating Systematic Relationships 1421 



tion! With every passing year electronic computers are becoming 
more efficient and more widely distributed. Thus the computational 
aspects of our method will become a progressively less important 
impediment. 

Since the matrix of correlation coefficients was unwieldy (it also 
had to be subdivided into sections ) and since further work with the 
correlation coefficients was contemplated, the latter were punched 
on 4656 IBM cards, one to a card. These cards were duplicated by 
means of the reproducing punch in order to obtain cards for a com- 
plete matrix of 9312 correlation coefficients. Information on these 
cards included matrix row and column numbers for the particular 
correlation, the coefficient with sign, and a class code for the co- 
efficient. These class code numbers (1-22) represented 22 classes 
of a frequency distribution of the correlation coefficients arrayed 
in ascending order of magnitude with class intervals of .05. In ad- 
dition, the cards contained codes for the relationship between the 
two species involved as evaluated by conventional systematic meth- 
ods (by CD. M.). 

The correlation coefficients on punched cards have so far been 
put to the following uses: We have compiled a printed tape record 
of the full matrix, column by column, which has been very useful 
for reference and further computation. Another tape has been com- 
piled giving a listing and frequency distribution of the correlation 
coefficients grouped in the 22 size classes. This tape has been of 
great value in various approaches to a classification of the relation- 
ships demonstrated by the matrix. A third tape lists the sums of 
the correlation coefficients, column by column. This has been nec- 
essary for the B-coefficient method briefly described below. A 
fourth tape presents a two-way frequency distribution showing the 
relation between correlation coefficients and the relationship code 
developed by conventional systematic methods. These tapes were 
prepared in a few hours running time from the correlation coefficient 





we still expect to use in a variety of ways. 

The matrix of correlation coefficients 

In the bee study the 4656 correlation coefficients computed in 
the above manner ranged in magnitude from — .0626 for the cor- 
relation between species 26 and 92, to .9747 for the correlation be- 
tween species 43 and 44. 6 As was mentioned previously, a fre- 

— . — i 

6. For lack of space the matrix cannot be reproduced here. Microfilm or IBM-tape 
or card copies can be obtained through the Secretary, Department of Entomology, Uni- 
versity of Kansas. Lawrence. 



1422 The University Science Bulletin 



quency distribution of these coefficients, grouped into 22 classes 
with class intervals of .05 was set up. The modal class showed a 
class mark of .38; this represents the most frequent class of cor- 
relation coefficients found between species in this study. How- 



ever, a second mode was located at .78. This bimodality would 
indicate that we are dealing with two populations of correlation 
coefficients: those indicating close, possibly intrageneric relations 



and others representing more distant relations. Codes representing 
Michener's previous views on the relationships among the species 
were correlated with the above coefficients. The single correla- 
tion coefficient between the correlation matrix and Michener's codes 
was .80. It was encouraging to find that magnitude of the corre- 
lation coefficients in our matrix was apparently an estimate of 
systematic relationship as indicated by the previous classification. 

Another way of examining these correlation coefficients is to study 
frequency distributions of the coefficients for any single species 
against all other species. By this means we were able to distinguish 
members of closely related groups of species from isolated species 
within a genus and these in turn from very isolated species represent- 
ing monotypic genera or subgenera. For a detailed discussion and 
illustrations of this procedure, the reader is referred to Miehener 
and Sokal ( 1957 ) . 

The absence of significant negative correlations from our matrix 



requires some discussion. Q-technique matrices of correlations be- 
tween people (based on psychological tests) are quite likely to 
yield such correlations. If there are distinct, antithetical types of 
persons represented in the matrix, such as extroverts and introverts, 
it is likely that a high score for one type will be a low score for the 



other and vice versa. In our case evolutionary progress may be 
represented by either an increase or a decrease in state codes. In 
the majority of characters the supposedly primitive situation is an 
intermediate state code with two diverging evolutionary trends rep- 
resented by the lower and higher code numbers. Furthermore, 
characters representing correlated trends were not necessarily coded 
along the same scale or in the same direction. It is clear that under 
such circumstances distantly related forms are likely to be uncor- 
related rather than negatively correlated. 

The search for group structure 

The matrix of correlation coefficients between species can be put 
to a variety of uses and the analysis reported below represents 
merely an initial effort at an exploitation of the data. The correla- 



Evaluating Systematic Relationships 1423 



» 



tion coefficients serve as an absolute measure of relationship be- 
tween any two species in our study, limited only insofar as the 
characters chosen do not represent the total correlated variation of 
the two species. 

The search for structure among the correlation coefficients of the 
matrix is of course no different in aim from the search by the sys- 
tematist for a natural system in an array of species. Such a system 
consists of a hierarchy of groups. Various methods can be used for 
discovering a hierarchy in data such as ours. A customary, rather 
simple device of the psychometrician is so-called "cluster analysis, 
developed to a fine art by Tryon ( 1939 ) . 

A concise description of the procedure (the ramifying linkage 
method) is given in Cattell (1944) and Thomson (1951). Because 
of the simplicity of the procedure, cluster methods are used exten- 
sively, although Cattell (1944, 1952) and others have pointed out 
that cluster analysis cannot be considered a substitute for the more 
involved factor analytic methods. Attempts to employ cluster anal- 
ysis for finding structure in our matrix were only partially success- 
ful, since the resulting clusters were partly overlapping, i. e., a given 
species might be a simultaneous member of two clusters. This 
makes good sense for intermediate forms in an abstract scheme of 
relationships. In a systematic hierarchic classification, however 
groups at the same level have to be mutually exclusive for practical 
as well as for theoretical reasons, except for low level groups exhibit- 
ing reticulate evolutionary pattern (rare above the species level in 
animals). A further reason for the unsuitability of cluster analysis 
is the complexity of the clusters as more species are added to them. 
Although clusters are therefore not convenient in an initial search 
for structure, the diagram of relationships established by methods 
to be described below could be easily recognized in the clusters 
outlined by cluster analysis. A method essentially similar to cluster 
analysis is the p- group and pF-group method of Olson and Miller 
(1951) applied to three paleontological R-technique matrices. It 
suffers from the same drawbacks as cluster analysis. 7 

7. After the present research and manuscript had been completed one of us (R. R. S.) 
became acquainted with the psychometric work of Professor Louis L. McQuitty of the 
Michigan State University, who in recent years has developed a whole battery of refined 
cluster methods (McQuitty, 1955, and a series of papers in press in The British Journal 
of Statistical Psychology, Educational and Psychological Measurement, and Psychological 
Monographs). Several of these papers deal with psychological problems which are closely 
related to those of biological classification. One of the methods invented by McQuitty 
bears a close resemblance to our variable group method developed below. It is interesting 
(as well as reassuring to us) that workers in different fields had unknown to each other 
developed some of the same formulations. We hope to try some of McQuitty's other methods 
on our material. They have the advantage of simplicity and can be programmed for elec- 
tronic computation without much difficulty. Indeed the time may not be far off when 
computation for a study such as our bee work will be a minor matter routinely handled by 
a computing center in a very few hours and the remaining problem will be the collection 
of data for the machine and the interpretation of the voluminous answers that are produced. 



9 




The University Science Bulletin 



As a technique for grouping the species we experimented exten- 
sively with the coefficient of belonging (B-coefficient) of Holzinger 
and Harman (1941). It is the sum of the correlations among the 
members of a group divided by the sum of the correlations of these 
group members with the other variables ( species ) of the study. 

Results of our B-coefficient analyses for the bees were reasonably 
good, as judged by the previous classification and by our subsequent 
investigations. There was one main drawback, however. Large 
species groups showed a lack of structure and relatively low B- 
coefficients which would make the species in these groups appear a 
good deal less related to one another than members of groups of 
two or three species. The cause of this phenomenon is not hard to 
find. In large species groups the denominator of the B-coefficient 
would include high correlations due to correlations of group mem- 
bers with numerous other prospective members not yet included in 
the group. This would tend to depress the B-coefficient values. By 
the time all such members have been admitted to the group, it has 
become so large that even the admission of a relatively unrelated 
variable will effect the B-coefficient only slightly. 

In view of the disadvantages of the B-coefficient we developed 
our procedure which is presented below in a general manner to- 
gether with some of the reasons for its adoption. This presentation 
is followed by a detailed step-by-step account of the computational 
procedure for readers who wish to become more familiar with it. 

A nucleus of a group was established, using the two species hav- 
ing the highest coefficient of correlation. Then species would be 
added to this nucleus, one at a time, always adding first the species 
having the highest average correlation with members of the group. 
The limit of the groups could be found by decreases ( L„ + x - L n ) 
in the level of the average correlation L n , where the subscript refers 
to the number of members in the group. As in the B-coefficient a 
significant drop is empirically determined since sampling distribu- 
tions of average correlations, such as L n , are unknown. By develop- 
ing first lower groups (species groups), then by the same method 



grouping these into larger groups (sometimes subgenera), and 
these into still larger or higher groups, etc., it has been possible to 
develop a hierarchy of groups for which the diagram of relationships 
( figure 1 ) can serve as a representative. Each number in this figure 
represents a different species; for a list of the species concerned see 
Michener and Sokal (1957). 

Since Ln is not amenable to rigorous statistical treatment it was 
decided to recompute correlation coefficients (using Spearman's sum 



Evaluating Systematic Relationships 



1425 



4 
C 



£ 



— b 



o 



00 

< 



a 

e 







f 



lr 



63 69 65 6T 70 75 77 83 78 79 94 85 88 90 92 
64 76 66 65 71 T2 82 81 74 80 73 «T 88 91 93 

I 



900 



U 



800- 



700- 








u 2 -° 

O .- 3 

"II — I 

94 96 97 
85 66 






l 
i 

f 







L 



600- 









ANTHOCOPA 




500 




PROTERIADES, 
20,30 HOPL1TI8, 40 & 41. 



\ 



\ 



^- 



400* 

Fig. 1. Diagram of relationships for the genus Ashmeadiella obtained by 
the weighted variable group method. Ordinate: magniture of correlation coef- 
ficient multiplied by 1000. Exact correlations between any two joining stems 
can be found by reading the value on the ordinate corresponding to the hori- 
zontal line connecting the stems. This value becomes approximate and maxi- 
mal in cases of multifid furcations. Broken lines used where more than three 
stems join are for convenience only; the horizontal connecting line has the same 
significance as elsewhere. "Roofs" over species numbers at the summits of the 
lines delimit subgenera containing more than one species, as based on C. D. M/s 
previous findings and not on this study. Generic names are in small capitals. 
The horizontal broken lines are not relevant to the present account; they are 
explained bv Michener and Sokal (1957). 



of variables method) after the group limits at each hierarchic level 
had been reached. Thus we returned at the end of each grouping 
procedure to a new matrix of correlation coefficients about which 
confidence statements might be made. Two further considerations 
in the final choice of a method for grouping remain to be mentioned: 
We might have admitted only one new member for each group 



at a given hierarchic level, thus obtaining a diagram of relationships 



consisting of bifurcations only. We have called this method the 



1426 



The University Science Bulletin 



pair-group method as contrasted with the variable-group method, 
where any number of new members can be admitted to the group at 
any one hierarchic level, the limit of the group being determined by 
a significant drop in L n . The pair-group method has some theoreti- 
cal justification in that much evolutionary ramification is believed 
based on speciational processes involving the splitting of one species 
into two. However, there must also occur some speciation as a re- 
sult of the splitting of a species into more than two isolates and, on 
the assumption of equal evolutionary rates for these new lines, the 
pair-groups method would fail to represent the true situation. More- 
over, many of the groups must be markedly different, not merely 
because of divergence, but because of extinctions of intermediates. 
A group might be broken into any number of different subgroups 
by different extinctions. Furthermore an empirical study of this 
method (see fig. 2 for an analysis of relationships in the subgenera 
Chilosima and Ashmeadiella by the pair-group method and com- 
pare with the left side of fig. 1 for the variable-group method) dem- 
onstrates that in spite of the pair-group device we are forced into 
multified furcations by drops in L n too small to plot or by temporary 



1000 



63 69 65 67 70 76 77 82 78 79 84 
64 75 66 68 71 72 83 81 74 80 85 



73 



900 



800 



700 




\ / 



T 




600 J 

Fig. 2. Diagram of relationships far the subgenera Chilosima (63-64) and 
Ashmeadiella $. str. ( 65-85 ) , obtained by the method of pair-groups, L e. dia- 
grams would ideally consist of bifurcations only. Stems have been weighted. 
Explanatory comments as for figure 1. 



Evaluating Systematic Relationships 




A 



B 



C 



D 



A 



B 



C 



D 



E 



F 



G 



H 




A , 




b. 



S 



T 



U 



V 



w 



X 



Y 



Z 




Fig. 3. Hypothetical diagrams of relationships to illustrate effects of different 

thods of weighting stems. For explanation see text. 



reversals of L n values, discussed below. Thus the variable-group 
method was adopted as the more reasonable and flexible of the two. 
A second consideration is how to weight the variables during the 
recalculation of the correlation matrix after each grouping pro- 
cedure. A simple diagram (fig. 3a) will make this issue clear. A 
and B represent the two species with the highest correlation coef- 
ficient. The L n for C against A and B is significantly below r ab , so 
that A and B are represented as being closer to each other than they 
are to C. When studying the relation of a fourth species D with 
group ABC we face the following problem: Should we calculate the 
correlation of ABC against D with A, B and C equally weighted or 
should we weight A = B and AB = C? Rephrased biologically, the 
problem is whether to relate species D with the homogeneous group 
ABC, or with the stem AB-C, where C carries as much weight in 



determining the relation with D as do A 






nd B together. 



Although 



in a simple case, such as the one described above the two alternatives 



may not 




uce very 




rent results, in a situation such as de- 



1428 The University Science Bulletin 



picted in fig. 3b species H might be weighted as % of the group A-H, 
or J2, depending on the system adopted. Similarly species B would 
be weighted % in the former case but only Yn in the latter case. 
When dealing with fairly large groups the second method would 
therefore reduce the weight of the early admitted members and in- 
crease the weight of those species admitted later. 

The same problem is found in a situation such as shown in fig. 
3c. By the first method species T is weighted 3b, by the second 
method it is weighted only H*. Neither of the two methods is en- 
tirely satisfactory. By method one we are reducing the importance 
of species H and Z in representing groups A-H and S-Z respectively. 
If the relationship diagrams of figures 3b and 3c depict true phylo- 
genetic relationships, then H and Z should represent half of their 
respective lines regardless of subsequent diversification in the other 
halves. On the other hand giving relatively greater weight to single 
late arrivals also gives heavier weight to specialized features of such 
species and thus would tend to distort the relational pattern, while 
specializations in the diversified branch of the stem tend to cancel 
each other, permitting a better average picture of the groups to 
emerge. The optimal system of weighting would be one between 
these two extremes, weighting each species according to its number 
of generalized and specialized features. This is clearly impossible 
without renewed introduction of a subjective element into our pro- 
cedure. We therefore adopted the second method, f. e., the weight- 
ing of new members as equal to the sum total of all old group mem- 
bers, thinking it to be the less objectionable of the two. We feel 
that this method will represent stems more correctly and that bias 
introduced by specializations of late joiners will be kept down by 
the large number of characters considered in our study. 

We are reassured in our decision by the results of a comparative 
study on the subgenera Chilosima and Ashmeadiella. Figure 4 
shows the results of a variable-group analysis of these subgenera by 
weighting method one, while results by method two can be seen in 
the left side of the diagram of fig. 1. General agreement as to re- 
lationships and level of furcations is very good. The main difference 
between the two diagrams is that in method one group 77-81 8 first 
receives 79 before receiving group 67-72, 78 and 84, while in method 



two it first receives group 67-72, then 78 and then 79 among others. 



■y 



8, In the interest of brevity groups will be identified by their leftmost (in the diagram) 
and rightmost members with a dash separating the two. Thus 77-81 means group 77, 82, 
83, 81. It clearly does not include all species ranging in number from 77 through 81, 
and includes some beyond that numerical range. 



Evaluating Systematic Relationships 



1429 



1000 



64 75 66 68 71 72 82 81 78 74 85 
63 69 65 67 70 76 77 83 79 94 80 73 



900 



800 








x 



I 



700 



X 



600 




(63-64) and 
ure under 



Fig. 4. Diagram of relationships for the subgenera Chilosima 
Ashmeadiella s. str. (65-85), obtained by the variable group 
weighting method one, i. e., equal weights for all stems. Explanatory com- 
ments as for figure 1. 



Careful examination of the original correlation coefficients makes 
the reasons for these differences clear. Group 77-81 is closer to 79 
than to 78 except that 81 is closer to 78 than to 79. Also 81 is closer 



to 67-72 than are 77, 




and 83. Therefore in method two, where 



81 receives as much weight as 77, 82 and 83 together, 67-72 joins 
the nuclear group first. This is also partly due to the fact that un- 
equal weighting of the species in the 67-72 group favors those close 
to the 77-81 group. Since 78 is closer to 67-72 than 79, the latter, 
while originally quite close to 77-81 is now temporarily delayed and 
78 joins the combined group 67-81 before 79 does. These relations 
are at too low a phyletic level to be included in the original diagram 
of relationships drawn by C. D. M. who feels that there is little that 
can be obtained from classical systematic studies of these species to 
suggest whether method one or method two is preferable. In view 



of the small over-all differences between the two methods and 
especially in view of the fact that the lines concerned all join by 
either method with a difference in correlation coefficients of less 
than .06, it may well be that we have made too great an issue of the 



matter. 



two 



will present us with a reasonably bias-free picture. 



1430 The University Science Bulletin 



The Weighted Variable Group Method 

It was thought advisable to give a detailed account of our method 
in order to enable readers to repeat the operations should they so 
desire. The subgenera Chilosima and Ashmeadiella, which have 
been used as a testing group before, will serve as an illustrative 
example. These subgenera include species 63 through 85 (see 

figure 1 ) . 

Correlation coefficients among these 23 species are shown in table 
4. All values are significant with probability values of less than one 
percent. The highest correlation coefficient among these species is 
.965 for 67 x 68. 9 This is also the highest correlation involving either 
of these two species. The next to enter group 67-68 is species 70 
which has the greatest average correlation ( L n = .892 ) with 67 and 
68 since 67 x 70 == .896 and 68 x 70 == .889. No other species in the 
study has as high an average correlation with 67-68, as can be 
learned from a few trials. We established empirically, as a result 
of numerous trials, that a drop in L n of .030 gave a satisfactory limit 
for groups; therefore 70 is not to be admitted to group 67-68 at this 



particular time. Another high correlation involving species other 
than 67 and 68 is 77 x 83 = .951. This is also the highest correlation 
for the two species concerned. Next to join this nucleus is species 
82 with an L n value of .936. The drop is less than .030; therefore 
82 is admitted. Next to join is species 79 with an L n value against 
77, 82 and 83 of .905. There is now a significant drop from the 
previous L n value and 79 is excluded for the time being. Drops in 
L n are always measured from the previous L n , not from the initial 
Ln. Our second group is therefore 77-83. In a similar manner we 
established groups 63-64, 65-66 and 71-76, each consisting of only 
two species. 

So far only 11 species out of the 23 of the study have been placed 
into groups. A systematic survey was then made of the remaining 
12 species to see if any group had been missed. For example, ex- 
amination of species 69 revealed that its highest correlation was 
with species 72 ( 69 x 72 = .820 ) . However, this latter value was not 
the highest correlation for 72, since 72 x 76 = .904. Thus 72 might 
eventually join the group containing 76, and 69 might join the group 
containing 72, both of which events came to pass at a later sta ( :e of 
the analysis. At the present time, however, species 69 and 72 are 

ed to any group. Similarly the remaining ten species 
in the study were shown not to belong to any nuclear group. To 





9. We shall use this symbolism in place of the more formal r^. 



OS 



Evaluating Systematic Relationships 



1431 



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1432 The University Science Bulletin 



set up one of the latter we required a correlation coefficient which 
was the highest one for both participating species (i.e., the re- 
ciprocally highest correlation). 

After the groups had been delimited a new correlation matrix 
was computed considering the newly formed groups as single 
variables, i.e., the previous matrix of 23 variables (matrix 1) was 
reduced to one of 17 variables (matrix 2). It is self-evident that 
the only correlation coefficients in need of recomputation were those 
involving new groups. Correlations involving only species that had 
remained single were not altered in any way. As a matter of fact, 
a procedure was devised by means of which the correlations were 
not even recopied, but variables joined into groups were crossed 
out and the new group variables were entered along the margins of 
the old matrix. The actual computational procedure is quite simple 
and considerably less complicated than the computations for finding 
the original correlation coefficients. It is described in the paper by 
its originator (Spearman, 1913) and also by Holzinger and Harman 
( 1941 ) . Let us illustrate this method by computing the correlation 
between groups (63) x and (67)i 10 . The general formula for this 
computation is 



DqQ 



• 



q.Q 



-^— ^^ 



— » 



V q + 2Aq V Q + 2AQ 



where □ qQ is the sum of all correlations between members of one 
group with the other group, Aq is the sum of all correlations be- 
tween members of the first group, AQ is a similar sum between 
members of the second group, q is the number of species in group 
one and Q the number of species in group 2. Thus in this particular 



case DqQ equals (63 x 67) + (63x68) + (64x67) + (64 x 68) 



= .692 + .682 + .681 + .672 = 2.727; Aq in this case equals only 
(63 x 64)= .90S while AQ equals 67 x 68 = .965, since each of these 
groups consists of two species only. In cases where a group con- 
sists of 3 species, for example, the A term consists of the sum of 
(1x2) -f- (1x3) + (2x3). In the present case q = Q = 2 species. 
Substituting into the formula given above: 



(63) 1 x(67) 1 



2.727 




V 2 + 2(.908) V 2 + 2(.965) 



10. The notation (63)i, refers to the group of species formed in matrix 1, the lowest 
numbered member of which is species 63, i.e., to group 63-64. Similarly (67)i, refers to 
67-68, and (77),, to 77-82-83. 



Evaluating Systematic Relationships 1433 



These computations can be set up in a systematic manner and are 
then neither particularly complicated nor time consuming. In the 
special case where we wish to calculate the correlation coefficient 
between a single species (x) and a new group (q), the formula is 
amended as follows: 



^ 



^ r X Mi 



*x*q 



V q + 2Aq 



An illustration is the correlation of species 69 with group (77 ) x . 2r x . Q 
equals ( 69x77 ) + ( 69x82 ) + ( 69x83 ) = .764 + .788 + .738 — 2.290, 
while Aq=(77x82) + (77x83) + (82x83)=.925+.951+.948=2. 




^ — ■ — —^—- 



2.290 



Then 69 x (77) x = = .779 



V 3 + 2(2.824) 

In such a manner a new 17 x 17 correlation matrix ( matrix 2 ) was 
constituted. From this point on the species groups [(63) 19 (65) a , 
(67) 1? (71) r , and (77) J were tested as though they were single 




Once matrix 2 had been computed the identical grouping pro- 
cedure was followed. Group (71) x had a mutually highest correla- 
tion withjpecies 70 at *923. They were then joined by 72 and group 
(67) x at L n levels of .903 and .885 respectively. These affiliations of 
72 and (67) 2 were also their highest correlations. The next prospec- 
tive joiner was species 81 at L n = .859, i. e. not quite the established 
drop of .030. However species 81 had highest relations not with the 
previous species but with group (77) x with a correlation of .923. 
Therefore it was excluded from consideration as a candidate for the 
earlier group and the runner up, species 78, used instead. The latter 
gave an L n value of .844, clearly a significant drop from .885. Species 



81 meanwhile was used in a nucleus of a new group 77-81 [(77) J. 
Situations such as the above were the exception. In general the re- 
lations and choices were entirely straightforward and could be left 
to the discretion of the computing assistants. 

At the end of each grouping procedure the remaining single 
variables were checked to avoid missing groups with low correla- 
tions between members. With each grouping procedure the matrix 
of correlation coefficients became smaller and the job of recomputa- 
tion less. The weighting procedure adopted by us was automatic 
in that all correlation coefficients used were from the previous matrix 
and not the initial one. It took eleven matrices to obtain a single 




The University Science Bulletin 



group out of the 23 species of the two subgenera. This amount of 
work could have been reduced by raising the minimum recognized 
difference in L n level above .030 but there would have been a re- 
sulting loss of detail in the diagram of relationships. Conversely, 
however, reducing the recognized difference below .030 would not 
have increased the meaningful detail, since even .030 was too small 
to prevent the occasional reversal of r values discussed below. 
Once computed, the relations were represented as diagrams of 
relationships as in figure 1. The ordinate at the left of each diagram 



is graduated in units of 1000 x r. The correlations between any join- 
ing stems in the diagram can be read by measuring the level along 
the ordinate of the horizontal line connecting the stems. Thus 
species 63 and 64 are correlated at a level of .908, while group 63-64 
is related to group 67-72 at .702. Furcations involving more than 
three lines are shown by broken lines converging on the midpoint 
of the horizontal line as in group 88-96 of the above figure. The 
tops of the figures are at a level of 1000 (correlation of 1) since 
obviously each species is perfectly correlated with itself. 

In cases of groups of only two stems the L„ level corresponds to 
the correlation coefficient of the two stems. When more than two 



stems join to form a group the highest L n level was graphed for all 
group members. Thus while 77 x 83 equals .951, L n for 82 against 
77 and 83 equals .936. The group of these three species (77-83) is 
shown related at levels .951. Occasionally the correlation coefficients 
for the same group in successive matrices will rise a little. Thus in 
this same figure groups 63-64, 69-85, 87-96, and species 86 are shown 



joining at .702. The first three groups actually joined at .671, but 



species 86 which joined their group at the next matrix did so at 
level .702. This type of situation, which occurred infrequently, might 
lead one to express concern about the validity of the method, since 



regular decreases in levels of correlation coefficients and L n values 
are expected. However, it can be shown from Spearman's formula 
for the correlation of sums of variables that slight increases in the 
levels of correlation coefficients of the sums of variables above the 



correlations of their component variables are possible. For ex- 
ample, if A and B have formed the nucleus of a group at r a . b = .9 
and G is about to join them, then by the rules of the variable group 
method both r a . c and r b . r must <r a . b = .9. It can then be shown 
that r (ab ). c must be <.925. Thus r ab . r , while it will usually be <r a . b , 
could be slightly more than .9. Similar situations can be shown to 
exist with larger-sized groups. The increases found by us were well 



Evaluating Systematic Relationships 




below the mathematically possible limits. In all such cases the re- 
lations were represented as multifid furcations of all the stems in- 
volved in the reversal and at the highest of the several L n levels con- 
sidered. 

In a successful method of studying relationships, the results of 
the analysis should be relatively independent of the number of 
species in the correlation matrix. If at least one species per species- 
group is included in a matrix, the ideal method of analysis should 
reproduce the diagram of relationships based on an earlier study of 
a larger matrix. If the method can be shown to produce similar 
results, the fact that our matrix contains only a sample from the 
population of species can be ignored with greater assurance. 

We tested this question by subjecting the odd-numbered species 
in the entire genus Ashmeadiella to a weighted variable group anal- 




. This should give an adequate cross-section of relationships in 
that genus. Since some trends might be lost by exclusion of the 



69 



65 



71 



83 



79 



85 



89 



93 



97 



1000 



63 



75 



67 



77 



81 



73 



87 



91 



95 



900 




I 



I 



r 



800 



700 



■r 



w 



% 



600 



m* 



Fig. 5. Diagram of relationships 
genus Ashmeadiella on the basis of 
planatory comments as for figure 1. 



predicted for odd numbered species of the 
the relationship diagram of figure 1. Ex- 



1436 



The University Science Bulletin 



even-numbered species a special diagram of relationships was pre- 
pared from figure 1 by using only odd-numbered species. Figure 5 
shows this predicted diagram of relationships. 

Figure 6 shows the results of the weighted variable group analysis 
on the odd-numbered species. There is less structure in this dia- 



69 



65 



71 



83 



79 



85 



89 



93 



97 



1000 



63 



75 



67 



77 



81 



73 



87 



91 



95 



900 



800 





t 



a 



\ 



700 




I 



I 



600 



an independent 
of the genus 



weighted 




Fig. 6. Diagram of relationships obtained by 
variable group analysis of the odd-numbered species 
Explanatory comments as for figure 1. 

gram as compared with the predicted diagram of Figure 5. This 
was to be expected since some of the structure was based on rela- 




g the missing 
tween the two 



In general, 



we therefore feel reassured that the species left out in our study 



g 



Evaluating Systematic Relationships 1437 




CONCLUDING COMMENTS 

A detailed discussion of the comparisons of our findings with the 

classifications of the four genera of bees is given by 
Michener and Sokal (1957). It will suffice here to state that general 
agreement was good but that a number of taxonomic re-evaluations 
seemed necessary as an outcome of these analyses. It should be re- 
membered that while diagrams such as figure 1 may suggest phyto- 
genies, in reality they only indicate static relationships. As indi- 
cated in the paper referred to, additional refinements were devised 
to give diagrams of relationships which we believe more nearly ap- 
proach phylogenetic trees. 

In view of these results we are encouraged to believe that, since 
the methods we have described are increasingly practical with the 
growing availability of high speed computers, this or similar schemes 
will be more widely utilized with different groups of organisms. 
Although the method we have described is a first attempt and would 
profit by either simplification or refinement, we believe it is a step 
toward reducing the subjectivity of systematic work, and therefore 
a step in the right direction. 

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59-96. 
Cattell, R. B. 

1944. A note on correlation clusters and cluster search methods. Psy- 

chometrika, vol. 9, pp. 169-184. 
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1954. Growing points in factor analysis. Austral. Journ. Psychol., vol. 6, 

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Holzinger, K. U., and H. H. Harman. 

1941. Factor analysis. Univ. of Chicago Press, Chicago, pp xii + 417. 

HOWELLS, W. W. 

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U1TTY, L. L. 

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The University Science Bulletin 



SOKAL 



1957. A quantitative approach to a problem in classification. Evolution, 

vol. 11, pp. 130-162. 
Olson, E. C, and R. L. Miller. 

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Evolution, vol. 5, pp. 325-338. 



SOKAL 



1952. Variation in a local population of Pemphigus. Evolution, vol. 6, 

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W 



ourn 



344-361. 



Stroud, C. P. 

1953. An application of factor analysis to the systematics of Kalotermes. 

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Nr Tryon, R. C. 




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